Abstract

On the basis of quantum-chemical calculations the most stable particle compositions are estimated in such model systems as (M+·[CrCl6] and M3CrCl6 + 18MCl (M = Na, K, and Cs). In all systems these particles are positively charged. For systems (M+·[CrCl6], (M+·[CrF6], M3CrF6 + 18MCl, M3CrF6 + 18MF, and M3CrCl6 + 18MCl (M = Na, K, and Cs) a number of energy parameters characterizing the state of the system before and after electron transfer are calculated. The results indicate the possibility of electron transfer from the cathode to the melt system, which is in the initial state. However, this possibility cannot be realized in systems where LUMOs (lowest unoccupied molecular orbitals) have purely ligand character. In this case, the preliminary deformation of a cationic shell of electroactive species is required; it transforms the initial system to the transition state. However, in all considered systems the search of the transition state should be carried close to the initial state . This greatly simplifies a problem and transforms it from a purely theoretical sphere to the field of practical tasks that do not require exceptional cost of computer time.

1. Introduction

Previously [1–4], quantum-chemical calculations of such model systems as (M+·[NbF7], (M+·[NbCl6], and (M+·[CrCl6] (M = Na, K, Cs; ) were carried out. These systems contain a complex anion with a cation outer-sphere (OS) shell. It was found that compositions with an intermediate number of OS cations , where is the limiting number of OS cations bound to a given complex, are thermodynamically most stable. Such systems (designating them as the type I system) are often used for modeling of the charge transfer process in melts [5–9]. The activation energy of electron transfer is estimated here based on Marcus theory [10]. These systems are suitable for express-evaluation of the charge transfer parameters, because it does not require large expenditures of computer time. However, it is clear that such simple systems can have only limited application. In particular, the effect of the anion composition of the electrolyte cannot be incorporated in the type I systems.

To verify the main findings for model systems I quantum-chemical computations of some parameters have been carried out in extended model systems of type II M3CrX6 + 18MX (M = Na, K, and Cs; X: F, Cl).

Some results for the CrF6-containing systems I and II were given previously [11–13]; for CrCl6-containing systems I and II calculations were made in the nonrelativistic basis sets for M: Na and K in the main [2, 4, 14, 15]. Here we provide data for the systems (M+·[CrCl6] and M3CrCl6 + 18MCl in a quasi-relativistic ECP basis for M: Na, K, and Cs and an additional analysis of all these systems.

2. Computational Methods

The quantum-chemical calculations were performed with the Firefly program package [16], partially based on the codes of the GAMESS (US) program [17], by the density functional theory methods (DFT/UHF). The spin-polarized version of the Kohn-Sham equations and the B3LYP hybrid functional were used. All calculations are made with the quasi-relativistic ECP basis set of Stuttgart/Dresden groups [18–20], more exactly, with Stuttgart RSC 1997 ECP for Cr, K, and Cs and Stuttgart RLC ECP for F, Cl, and Na. In addition Na-systems in some cases used CRENBL ECP basis set [20]. The energy values correspond to true minima of the potential energy surface (the imaginary frequencies are absent). In all complexes chromium is in a high-spin state: for Cr(III) and for Cr(II). Spin contamination can be ignored: for the Cr(III) and Cr(II) particles, the values were 3.76–3.82 and 6.01–6.03, respectively (3.75 and 6.00 in the ideal case).

3. Results and Discussion

3.1. Composition of the Most Stable Particles

In systems II calculation of the interaction energy of the second coordination sphere fragments (M+ with both the complex and the external environment of a given fragment can be made. Figures 1(a) and 1(b) show the plots of the energy of cationic OS shell formation in systems I (M+·[CrCl6]3− (Figure 1(a)) and II M3CrCl6 + 18MCl (Figure 1(b)) calculated by where , , and are the energies of systems I (or an equivalent fragment of systems II), free [CrCl6]3− complex, and free M+ cation, respectively. The energy value reflects the total effect of the interaction of OS cations M+ with the complex [CrCl6]3− and with each other.

Dependence of this type always has a minimum at some intermediate value. The existence of the minimum is mainly caused by an increase in the repulsion between OS cations as their number increases. The composition of the system at point is the most stable. In the case of systems I for all M (Figure 1(a)). The maximal number of OS cations retained by the chromium complex in systems I is 6.

Figure 1(b) shows the energy of the OS shell formation for analogous fragments of systems II (M+·[CrCl6]3−. Note, in systems II, the OS cations are ranked by increasing the (Cr-M) distances (); that is, with increase of the number on Figure 1(b) the distance (Cr-) increases, too. The second coordination sphere of chromium in these systems contains 8-9 cations; however the minimum corresponds to equal to 4 (Na, K) or 5 (Cs) only. Figure 2 shows the examples of model structures for systems I and II.

Thus, no significant displacement of the minimum was observed during the transition from systems I to systems II. This allows considering data based on results for systems I as a valid initial estimate.

Unlike the systems I, for the systems II the interaction energy of the second coordination sphere (M with the complex and the rest of the system can be calculated directly. If the interaction energy of this shell with the complex exceeds the energy of its interaction with the outer environment, we can state that there is a dynamic equilibrium in the system responsible for the existence of rather stable complex species of definite composition [11]. Relevant data are presented below.

Figure 3 shows the energy of interaction of the (M+·[CrCl6]3− fragment with the outer environment (the rest of the systems II) as a function of the OS cations number. The values were calculated from the following equation:

Figure 3: Interaction energy (kJ/mol) of the (M+⋅[CrCl6]3− species in systems II M3CrCl6 + 18MCl versus the number of OS cations for (1) Na, (2) K, and (3) Cs.

Here the symbols , , and denote energies of the entire model system, its fragment (Mn)·[CrCl6]3−, and the rest of systems II, respectively.

The maximal value of this energy is observed at . The compositions closest in energy to the composition with the minimum energy are given in parentheses. Thus, the composition of the fragment (M+·[CrCl6]3− at which its interaction with the environment is minimal () is close to the most stable compositions mentioned in Figure 3 (, ).

Finally, let us consider the function ΔE versus in Figure 4. It was obtained in the following way. First, two types of the interaction energy of the cation OS shell (M+ were calculated: (i) the energy of interaction with the complex and (ii) the energy of interaction with the rest of systems II . The ΔE value in Figure 4 is equal to the difference:

Figure 4: The ΔE value (kJ/mol) in systems M3CrCl6 + 18MCl versus the number of OS cations for () Na, () K, and () Cs. Explanation in the text.

If we denote such fragments as [CrCl6], (Mn), and the rest of system by characters , , and , respectively, then the first energy corresponds to equation . The second energy, that is, , characterizes the equation , while the energy ΔE corresponds to equilibrium .

As follows from Figure 4, in the range –6, the energy of OS shell interaction (M+· with the complex is larger than with the external rest of system in all model systems. Maximum of this function corresponds to the value .

Thus, the most stable particles in systems II are (M+·[CrCl6]3− with (Na, K) or 5 (Cs), as follows from Figure 1(b). According to Figure 4, such fragments can be considered as a single particle relatively weakly bound to the surroundings; they have a positive charge for all M.

3.2. Comparison of the System Energies before and after Electron Transfer

Let us introduce the following notation. is initial state of the system (before electron transfer); is final state of the system (after electron transfer); is state of the system with the geometric structure , but after the electron transfer (ET); is state of the system with the geometric structure , but before ET; is the energy of system in state ; ; ; . Electron transfer here means the transfer of one electron in the cathode process to the complex of chromium: [CrX6]3− + ē → [CrX6]4− (X – F, Cl).

Table 1: The energy parameters (kJ/mol) of the systems before and after electron transfer. Explanation in text.

Here we should pay attention to the following. In the range of –4, the energy values ΔE2 and ΔE3 change the sign in the system (M+·[CrF6]3−. Therefore, in this range of the n value mechanism of ET will depend on the composition of the electroactive particle. In particular for Na-system at there is intermediate structure state in which the intersection of the potential energy surfaces of states, before and after ET, occurs. This fact is the basis for estimation of the ET activation energy according to the Marcus theory [10]. If composition of the electroactive species in the Na-system is close to the one with equal 4, then ET may occur directly to the complex in the initial state (in case when there are no other restrictions). These considerations are also applicable to K-, Cs-systems.

In the system (M+·[CrCl6]3− only functions ΔE2 change the sign. However, this is quite enough to use for states with and 4, the conclusions obtained above in relation to the initial state . Low stability of the state in comparison with the state is due to the strong influence of the Jahn-Teller effect on the structure of the [CrCl6]4− complex after ET. For this reason, the state is strongly destabilized and the ΔE3 energy is less than zero for all values of .

In the extended systems II the ΔE values are less than zero for all M. In this case, formally electron transfer can occur at the initial state . Some comments on this conclusion are made in the next section.

4. Conclusions

Results of this work indicate the possibility of ET directly to the initial particle , rather than through a transition state . This finding needs to be clarified.

Earlier [21] analysis of the nature of frontier molecular orbitals was held in systems M3CrX6 + 18MCl (X: F, Cl; M: Na, K, and Cs) and it was shown that ET through the OS cations is only possible in system Cs3CrF6 + 18CsCl. In other systems, the nature of the LUMO (lowest unoccupied molecular orbital) requires preliminary adjustment of the OS shell to provide direct contact ligands of the first coordination sphere with the electrode surface. That is, in these systems, ET has to take place in a transition state .

In work [21] it was also suggested that for the system Cs3CrF6 + 18CsCl charge transfer takes place in a state with geometric structure which is close to the structure of the initial system. In this work we obtain confirmation of this assumption.

In the Cs3CrF6 + 18CsCl system ET occurs with abnormally high rate and apparently is limited by diffusion. The approach of the active particles to the electrode surface requires the overcoming of the activation barrier, which depends on the particle charge. Data related to the composition and the possible charge of such particles can be obtained from data such as in Figures 1, 3, and 4. An example of use of such approach is contained in the work [22] where the abnormal rate of ET in the system Na2NbCl6 + 18MCl in comparison with systems M2NbCl6 + 18MCl (M – K, Cs) has been explained by distinction of charge of the electroactive species.

Furthermore, according to calculations, in some cases a higher rate of ET can occur for particles with less stability [13].

Thus, analysis of the mechanism of electrochemical ET in melts requires taking into account a number of factors. Nevertheless, the main conclusion of this work remains true: the search of the transition state in the considered systems should be carried close to the initial state . This greatly simplifies a problem and transforms it from a purely theoretical sphere to the field of practical tasks that do not require exceptional cost of computer time.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the Russian Foundation for Basic Research (Project no. 15-03-02290-a).